Obtain the following formula without attempting to justify the steps used in the process.
![\[ \sum_{n=0}^{\infty} (n+1)x^n = \frac{1}{(1-x)^2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-b6aab38fd0740f1f28365f2f2e96e725_l3.png "Rendered by QuickLaTeX.com")
Starting with the formula for the sum of a geometric series and differentiating both sides (without justifying that we can do this) we have
![\begin{align*} &&\sum_{n=0}^{\infty} x^n &= \frac{1}{1-x} \\[9pt] \implies && \sum_{n=0}^{\infty} nx^{n-1} &= \frac{1}{(1-x)^2} \\[9pt] \implies && \sum_{n=-1}^{\infty} (n+1)x^n &= \frac{1}{(1-x)^2} \\[9pt] \implies && \sum_{n=0}^{\infty} (n+1)x^n &= \frac{1}{(1-x)^2}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7d6cf6a3eadd15f343867b7e9474d0c1_l3.png "Rendered by QuickLaTeX.com")
Obtain the following formula without attempting to justify the steps used in the process.
Starting with the formula for the sum of a geometric series and differentiating both sides (without justifying that we can do this) we have
Why is it that only 1 of the n’s in the last line is incremented?
Also, this is doable by turning (n+1)x^n into nx^n + x^n which is exercise 11 + the sum of a geometric series